On a K-1 visa, your birth certificate is not required until the end of the process when you go to your medical exam and your interview at the U.S. Embassy.
You have some time to correct your birth certificate with your local civil office. It’s worth it to take care of this now because if you get to the interview and the consular officer issues what is called a request for evidence (RFE) for a new birth certificate, it could take some time and delay your case.
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